Self-immittance refers to either impedance, or its reciprocal, admittance, measured between a single pair of terminals. At a given discrete frequency, self-immittance is a complex quantity. As such, it contains two components and can be expressed as either magnitude and phase, or real and imaginary parts. These two alternative forms of complex self-impedance/admittance are equivalent.
Patents have recently been applied for by Champlin which are directed to a technique for measuring complex self-immittance of electrochemical cells and batteries (see U.S. Pat. No. 6,002,238, U.S. patent application Ser. No. 09/454,629, and U.S. patent application Ser. No. 09/503,015). Special provisions of that technique take into account the very low impedance level and the intrinsic internal voltage of a cell or battery. One aspect of the present invention includes measuring complex self-impedance/admittance of a general two-terminal electrical element having arbitrary impedance and that may, or may not, contain an internal voltage source.
Balanced bridge methods for measuring complex self-immittance at a discrete frequency are well known. Two examples are shown in FIGS. 1a and 1b. FIG. 1a illustrates a bridge circuit for directly measuring real and imaginary parts of the complex self-impedance, Z=R+jX, of a general element denoted GE, where j=-1+L . FIG. 1b illustrates a bridge for directly measuring a general element's real and imaginary parts of self-admittance, Y=G+jB. Although complex Z and Y are related by the reciprocal relationship Y=1/Z, the same is not true of their real and imaginary components (G.noteq.1/R, B.noteq.1/X). Hence, the distinctly different bridge circuits.
Consider either FIG. 1a or FIG. 1b. At a given frequency of the sinusoidal generator, one adjusts the calibrated variable resistance or conductance element, R.sub.X or G.sub.X, and the calibrated variable reactance or susceptance element, X.sub.X or B.sub.X, for bridge balance as indicated by the null detector. By virtue of the equal resistors R.sub.B on either side of the bridge, the balance condition indicates that R.sub.X =R and X.sub.X =X (FIG. 1a) or G.sub.X =G and B.sub.X =B (FIG. 1b). Accordingly, the real and imaginary parts of the unknown self-immittance, either Z=R+jX or Y=G+jB, an be read directly from the calibrated values of the appropriate two variable elements.
Bridge methods for measuring complex self-immittance suffer from several disadvantages. First of all, obtaining an accurate balance condition is a very time-consuming procedure that generally requires exceptional skill. Secondly, bridge accuracy is critically dependent upon the calibration accuracy of the variable elements. Finally, calibrated reactance and susceptance elements that are variable over a wide adjustment range are very difficult to implement.
A second prior-art technique for measuring complex self-immittance of a general element at a particular discrete frequency is illustrated in FIG. 2. In this circuit, a sinusoidal current i(t) excites the unknown element. This excitation current is sensed across a "viewing" resistor, which, for simplicity, is assumed to be 1 ohm. Accordingly, in this example, the sensed current-signal voltage across the resistor is numerically equal to i(t). This excitation current signal is presented to the horizontal input of an oscilloscope, and the responding voltage signal across the general element, v(t), is presented to the vertical input. The resulting display is known as a single-loop Lissajous' pattern.
If, for simplicity, the horizontal and vertical gains are chosen to be equal, one can determine the magnitude and phase of the unknown self-impedance directly from the displayed Lissajous' pattern. The magnitude is simply the ratio of maximum vertical excursion to maximum horizontal excursion, and the phase angle is the inverse sine of the ratio of zero-crossing point to maximum excursion. Although this Lissajous' pattern technique is very simple, it is not particularly accurate since it depends critically upon the operator's visual acuity.
A third prior-art method for measuring discrete-frequency complex self-admittance of a general element--a method that is closely related to the present invention--is illustrated in FIG. 3. This technique utilizes apparatus, often referred to as a "frequency response analyzer" (FRA), implementing a measuring technique known as "sine/cosine correlation".
Consider FIG. 3. A sinusoidal generator generates zero-phase reference voltage v(t)=V sin(.omega.t). Because of the feedback-induced "virtual short-circuit" at the input of the operational amplifier, reference voltage v(t) appears directly across the unknown GE, and thus serves as its excitation. The responding current through the GE can be written in the form i(t)=I.sup.0 sin(.omega.t)+I.sup.90 cos(.omega.t) where I.sup.0 is the amplitude of the current component that is in time-phase with the reference voltage, and I.sup.90 is the amplitude of the component in time-quadrature. Assuming for simplicity that the feedback resistor is 1 ohm, the voltage signal at the output of the operational amplifier is numerically equal to -i(t). Thus, the operational amplifier serves as a current-to-voltage converter.
The two signals, v(t) and -i(t), are multiplied together in a first hardware multiplier. The multiplier's output is the product -VI.sup.0 sin.sup.2 (.omega.t)-VI.sup.90 sin(.omega.t)cos(.omega.t), which, by using well-known trigonometric identities, can be written -(VI.sup.0 /2)+(VI.sup.0 /2)cos(2.omega.t)-(VI.sup.90 /2)sin(2.omega.t). Integrating this signal with a first hardware integrator (low-pass filter) removes the two time-varying components leaving only the dc voltage -(VI.sup.0 /2).
The two signals at the inputs of the second multiplier are -i(t) and a signal V cos(.omega.t) obtained by shifting v(t) in time-phase by 90.degree.. Again by using well-known trigonometric identities, the multiplier's output can be shown to be -(VI/2)sin(2.omega.t)-(VI.sup.90 /2)-(VI.sup.90 /2)cos(2.omega.t). Integrating this signal with a second hardware integrator removes the two time-varying components leaving only the dc voltage -(VI.sup.90 /2).
Both inputs of the third multiplier are v(t)=V sin(.omega.t). The output signal is therefore V.sup.2 sin.sup.2 (.omega.t) which, by using a trigonometric identity, can be shown to be equivalent to (V.sup.2 /2)-(V.sup.2 /2)cos(2.omega.t). Integrating this signal with a third hardware integrator removes the time-varying component leaving only the dc voltage (V.sup.2 /2).
Finally, the dc outputs of the first and second integrators are divided by the dc output of the third integrator. These two divisions yield -G=(-I.sup.0 /V) and -B=(-I.sup.90 /V), respectively, the negative real and imaginary parts of admittance Y=G+jB of the unknown element. Thus, by employing a zero-phase reference voltage as excitation and sensing the resulting in-phase and quadrature components of current response, the apparatus of FIG. 3 fundamentally measures components of complex self-admittance. One sees further that this technique employs hardware devices to evaluate the two correlation integrals .intg.i(t)sin(.omega.t)dt and .intg.i(t)cos(.omega.t)dt. Hence the name "sine/cosine correlation".
Improvements and variations on this basic "sine/cosine correlation" technique have been described by Jackson in U.S. Pat. No. 3,808,526; by Allison in U.S. Pat. No. 4,322,806; by Sugihara in U.S. Pat. No. 4,409,543; by Ryder in U.S. Pat. No. 4,868,487; by Wakasugi, et al., in U.S. Pat. No. 4,888,701; by Kitayoshi in U.S. Pat. No. 4,947,130; by Wakasugi in U.S. Pat. No. 4,935,692; and by Park in U.S. Pat. No. 5,519,325.